ISEN 609 Lecture 9 - Renewal Processes Suppose cfw_N(t t 0...

Renewal ProcessesSuppose is a counting process where the times between arrivals of events are independent, identically distributed random variables.In this case, we do not requirethat the times between events be exponentially distributed.Let Fbe the distribution function of the times between events, and suppose the mean of this d.f. is and the variance is(historically, “events” are referred to as “renewals” because at times of events, the process starts over, or “renews” itself){N(t), t≥0}μσ2

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Computing ConvolutionsSuppose the interrenewal times are exponentially distributedYou should recognize this as the gamma cdf. P(S2≤t) =P(X1+X2≤t)=t0P(X1+X2≤t|X2=u)dF(u)=t0P(X1≤t-u)dF(u)=t0(1-e-λ(t-u))λe-λudu=t0λe-λudt+t0λe-λtdt= 1-e-λt+e-λtλt

Computing ConvolutionsSuppose the interrenewal times are uniformly distributed on [0, b]Note we can also do this using the mgf!P(S2≤t) =P(X1+X2≤t)=t0P(X1+X2≤t|X2=u)dF(u)=t0P(X1≤t-u)dF(u)=t0(t-u)b1bdu=t22b2

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